Search results for 'Number theory' (try it on Scholar)

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  1.  16
    Renling Jin (2000). Applications of Nonstandard Analysis in Additive Number Theory. Bulletin of Symbolic Logic 6 (3):331-341.
    This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.
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  2.  1
    Sue Ann Toledo (1975). Tableau Systems for First Order Number Theory and Certain Higher Order Theories. Springer-Verlag.
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  3.  60
    Jeremy Avigad (2003). Number Theory and Elementary Arithmetic. Philosophia Mathematica 11 (3):257-284.
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  4.  30
    J. Ellenberg & E. Sober (2011). Objective Probabilities in Number Theory. Philosophia Mathematica 19 (3):308-322.
    Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in (...)
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  5.  27
    J. Michael Dunn (1979). A Theorem in 3-Valued Model Theory with Connections to Number Theory, Type Theory, and Relevant Logic. Studia Logica 38 (2):149 - 169.
    Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (...)
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  6.  1
    Alessandro Berarducci & Benedetto Intrigila (1991). Combinatorial Principles in Elementary Number Theory. Annals of Pure and Applied Logic 55 (1):35-50.
    We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give (...)
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  7. C. Pomerance (2008). Computational Number Theory. In T. Gowers (ed.), Princeton Companion to Mathematics. Princeton University Press 348--362.
     
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  8.  32
    Marcus Alfred, Petero Kwizera, James V. Lindesay & H. Pierre Noyes (2004). A Nonperturbative, Finite Particle Number Approach to Relativistic Scattering Theory. Foundations of Physics 34 (4):581-616.
    We present integral equations for the scattering amplitudes of three scalar particles, using the Faddeev channel decomposition, which can be readily extended to any finite number of particles of any helicity. The solution of these equations, which have been demonstrated to be calculable, provide a nonperturbative way of obtaining relativistic scattering amplitudes for any finite number of particles that are Lorentz invariant, unitary, cluster decomposable and reduce unambiguously in the nonrelativistic limit to the nonrelativistic Faddeev equations. The aim (...)
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  9.  18
    Toby Ord (2007). Ω in Number Theory. In C. S. Calude (ed.), Randomness and Complexity, from Leibniz to Chaitin. World Scientific 161-173.
    We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be (...)
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  10.  21
    Christopher Norris (2002). Putnam, Peano, and the Malin Génie: Could We Possibly Bewrong About Elementary Number-Theory? [REVIEW] Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 33 (2):289-321.
    This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also (...)
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  11. Manfred Szabo (1970). On the Original Gentzen Consistency Proof for Number Theory. In A. Kino, John Myhill & Richard Eugene Vesley (eds.), Intuitionism and Proof Theory. Amsterdam,North-Holland Pub. Co. 409.
     
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  12. Colin Mclarty (2010). What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory. Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  13. Steven Orey (1955). Formal Development of Ordinal Number Theory. Journal of Symbolic Logic 20 (1):95-104.
  14. T. M. Scanlon (1973). The Consistency of Number Theory Via Herbrand's Theorem. Journal of Symbolic Logic 38 (1):29-58.
  15.  67
    S. C. Kleene (1945). On the Interpretation of Intuitionistic Number Theory. Journal of Symbolic Logic 10 (4):109-124.
  16.  5
    Wolfram Pohlers (1998). Subsystems of Set Theory and Second Order Number Theory. In Samuel R. Buss (ed.), Bulletin of Symbolic Logic. Elsevier 137--209.
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  17.  6
    Alonzo Church (1936). An Unsolvable Problem of Elementary Number Theory. Journal of Symbolic Logic 1 (2):73-74.
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  18.  5
    J. C. Shepherdson (1965). A Non-Standard Model for a Free Variable Fragment of Number Theory. Journal of Symbolic Logic 30 (3):389-390.
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  19.  4
    Andrew Adler (1969). Extensions of Non-Standard Models of Number Theory. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (19):289-290.
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  20.  4
    Albert A. Mullin (1965). A Contribution Toward Computable Number Theory. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 11 (2):117-119.
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  21.  4
    Samuel T. Stern (1969). A Number Theory for the Seminaturals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (26-29):401-410.
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  22.  8
    Alan C. Bowen (2008). Boethian Number Theory. Ancient Philosophy 9 (1):137 - 143.
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  23.  27
    G. Kreisel (1952). On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications. Journal of Symbolic Logic 17 (1):43-58.
  24.  34
    Hilary Putnam (1960). An Unsolvable Problem in Number Theory. Journal of Symbolic Logic 25 (3):220-232.
  25.  22
    B. Mazur (1994). Questions of Decidability and Undecidability in Number Theory. Journal of Symbolic Logic 59 (2):353-371.
  26.  21
    Alan C. Bowen (1989). Boethian Number Theory: A Translation of the de Institutione Arithmetica with Introduction and Notes. Ancient Philosophy 9 (1):137-143.
  27.  25
    Yvon Gauthier (2008). From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory Paolo Bussotti Augsburg, Erwin Rauner Verlag, 2006, 574 p. [REVIEW] Dialogue 47 (2):411.
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  28.  28
    R. L. Goodstein (1947). Transfinite Ordinals in Recursive Number Theory. Journal of Symbolic Logic 12 (4):123-129.
  29.  3
    Toshiyasu Arai (2000). Pohlers Wolfram. Subsystems of Set Theory and Second-Order Number Theory. Handbook of Proof Theory, Edited by Buss Samuel R., Studies in Logic and the Foundations of Mathematics, Vol. 137, Elsevier, Amsterdam Etc. 1998, Pp. 209–335. [REVIEW] Bulletin of Symbolic Logic 6 (4):467-469.
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  30.  23
    Yvon Gauthier (1978). Foundational Problems of Number Theory. Notre Dame Journal of Formal Logic 19 (1):92-100.
  31.  12
    Steven C. Leth (1988). Some Nonstandard Methods in Combinatorial Number Theory. Studia Logica 47 (3):265 - 278.
    A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.
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  32.  17
    W. Knorr (1976). Problems in the Interpretation of Greek Number Theory: Euclid and the 'Fundamental Theorem of Arithmetic'. Studies in History and Philosophy of Science Part A 7 (4):353-368.
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  33.  16
    Jeremy Avigad, Kevin Donnelly, David Gray & Adam Kramer, Number Theory.
    1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...)
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  34.  4
    Elliott Mendelson (1961). On Non-Standard Models for Number Theory. In Bar-Hillel, Yehoshua & [From Old Catalog] (eds.), Journal of Symbolic Logic. Jerusalem, Magnes Press, Hebrew University; 259--268.
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  35.  6
    Martin Davis (1990). Gödel Kurt. Zur Intuitionistischen Arithmetik Und Zahlentheorie (1933e). A Reprint of 41811. Collected Works, Volume I, Publications 1929–1936, by Kurt Gödel, Edited by Feferman Solomon, Dawson John W. Jr., Kleene Stephen C., Moore Gregory H., Solovay Robert M., and van Heijenoort Jean, Clarendon Press, Oxford University Press, New York and Oxford 1986, Even Pp. 286–294. Gödel Kurt. On Intuitionistic Arithmetic and Number Theory (1933e). English Translation by Stefan Bauer-Mengelberg and Jean van ... [REVIEW] Journal of Symbolic Logic 55 (1):346-346.
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  36.  6
    Wilbur Knorr (1985). Boethius, Boethian Number Theory: A Translation of the “De Institutione Arithmetica,” Trans. Michael Masi. (Studies in Classical Antiquity, 6.) Amsterdam: Rodopi, 1983. Paper. Pp. 197; 6 Illustrations. $27.75. Distributed in the U.S.A. By Humanities Press, Atlantic Highlands, N.J. [REVIEW] Speculum 60 (4):946-948.
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  37. William Craig (1953). Review: John Myhill, A Derivation of Number Theory From Ancestral Theory. [REVIEW] Journal of Symbolic Logic 18 (1):77-77.
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  38.  16
    W. D. Goldfarb & T. M. Scanlon (1974). The Ω-Consistency of Number Theory Via Herbrand's Theorem. Journal of Symbolic Logic 39 (4):678-692.
  39. R. L. Goodstein (1958). Recursive Number Theory. A Development of Recursive Arithmetic in a Logic-Free Equation Calculus. Journal of Symbolic Logic 23 (2):227-228.
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  40.  5
    John Myhill (1952). A Derivation of Number Theory From Ancestral Theory. Journal of Symbolic Logic 17 (3):192-197.
  41.  10
    F. G. Asenjo (1965). The Arithmetic of the Term-Relation Number Theory. Notre Dame Journal of Formal Logic 6 (3):223-228.
  42.  9
    Dorothy Bollman (1967). Formal Nonassociative Number Theory. Notre Dame Journal of Formal Logic 8 (1-2):9-16.
  43.  9
    Robert R. Tompkins (1968). On Kleene's Recursive Realizability as an Interpretation for Intuitionistic Elementary Number Theory. Notre Dame Journal of Formal Logic 9 (4):289-293.
  44.  7
    Nino Cocchiarella (1984). Formal Number Theory and Compatibility. [REVIEW] Teaching Philosophy 7 (4):361-362.
  45.  2
    Wilbur Knorr (1985). Boethian Number Theory: A Translation of the “De Institutione Arithmetica,”. [REVIEW] Speculum 60 (4):946-948.
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  46.  3
    Andrew Adler (1969). Extensions of Non‐Standard Models of Number Theory. Mathematical Logic Quarterly 15 (19):289-290.
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  47.  8
    Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.
  48.  3
    Th Skolem (1958). Review: R. L. Goodstein, Recursive Number Theory. A Development of Recursive Arithmetic in a Logic-Free Equation Calculus. [REVIEW] Journal of Symbolic Logic 23 (2):227-228.
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  49.  5
    Albert A. Mullin (1961). Correlative Remarks Concerning Elementary Number Theory, Groups and Mutant Sets. Notre Dame Journal of Formal Logic 2 (4):253-254.
  50.  6
    Rosina Albano- Zinco (1975). On Gurwitsch's Number Theory. Graduate Faculty Philosophy Journal 5 (1):109-112.
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